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Theorem frgrwopreglem5lem 29553
Description: Lemma for frgrwopreglem5 29554. (Contributed by AV, 5-Feb-2022.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
frgrwopreg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrwopreglem5lem (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝐴,𝑏   𝑥,𝐵   𝑦,𝐷   𝐺,𝑎,𝑏,𝑦,𝑥   𝑦,𝑉
Allowed substitution hints:   𝐴(𝑦,𝑎)   𝐵(𝑦,𝑎,𝑏)   𝐷(𝑎,𝑏)   𝐸(𝑥,𝑦,𝑎,𝑏)   𝐾(𝑦,𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem frgrwopreglem5lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
21reqabi 3455 . . . . 5 (𝑥𝐴 ↔ (𝑥𝑉 ∧ (𝐷𝑥) = 𝐾))
3 fveqeq2 6897 . . . . . . 7 (𝑥 = 𝑎 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑎) = 𝐾))
43, 1elrab2 3685 . . . . . 6 (𝑎𝐴 ↔ (𝑎𝑉 ∧ (𝐷𝑎) = 𝐾))
5 eqtr3 2759 . . . . . . . . 9 (((𝐷𝑎) = 𝐾 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥))
65expcom 415 . . . . . . . 8 ((𝐷𝑥) = 𝐾 → ((𝐷𝑎) = 𝐾 → (𝐷𝑎) = (𝐷𝑥)))
76adantl 483 . . . . . . 7 ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → ((𝐷𝑎) = 𝐾 → (𝐷𝑎) = (𝐷𝑥)))
87com12 32 . . . . . 6 ((𝐷𝑎) = 𝐾 → ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥)))
94, 8simplbiim 506 . . . . 5 (𝑎𝐴 → ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥)))
102, 9biimtrid 241 . . . 4 (𝑎𝐴 → (𝑥𝐴 → (𝐷𝑎) = (𝐷𝑥)))
1110imp 408 . . 3 ((𝑎𝐴𝑥𝐴) → (𝐷𝑎) = (𝐷𝑥))
1211adantr 482 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) = (𝐷𝑥))
13 frgrwopreg.v . . . 4 𝑉 = (Vtx‘𝐺)
14 frgrwopreg.d . . . 4 𝐷 = (VtxDeg‘𝐺)
15 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
1613, 14, 1, 15frgrwopreglem3 29547 . . 3 ((𝑎𝐴𝑏𝐵) → (𝐷𝑎) ≠ (𝐷𝑏))
1716ad2ant2r 746 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) ≠ (𝐷𝑏))
18 fveqeq2 6897 . . . . . 6 (𝑥 = 𝑧 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑧) = 𝐾))
1918cbvrabv 3443 . . . . 5 {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
201, 19eqtri 2761 . . . 4 𝐴 = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
2113, 14, 20, 15frgrwopreglem3 29547 . . 3 ((𝑥𝐴𝑦𝐵) → (𝐷𝑥) ≠ (𝐷𝑦))
2221ad2ant2l 745 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑥) ≠ (𝐷𝑦))
2312, 17, 223jca 1129 1 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  {crab 3433  cdif 3944  cfv 6540  Vtxcvtx 28236  Edgcedg 28287  VtxDegcvtxdg 28702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548
This theorem is referenced by:  frgrwopreglem5  29554
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